Complex Trigonometric Functions Research Articles

Molecular dynamics study of bubble nucleation on trigonometric nanostructured surfaces

Nanostructures have a crucial impact on bubble nucleation. In this paper, different nano-grooved surfaces are constructed using complex trigonometric functions, and the nucleation process of liquid argon on the surfaces is investigated using molecular dynamics simulations. The nanostructures affect the bubble nucleation location, and the results show that the bubble nucleation always occurs in the groove region during the boiling process. In contrast, the bubbles on the flat surface appear randomly. In addition, bubble growth is also affected by the nanostructure. Increasing nanostructure size can reduce the bubble nucleation time and enhance the bubble growth rate. The nanostructure increases the solid-liquid interface area, which enhances the heat transfer between solid and liquid. Finally, the reasons for bubble nucleation on different nanostructured surfaces are analyzed. The heat transfer between solid and liquid is the leading factor for bubble nucleation when the difference in liquid atoms’ potential energy is slight. As the nanostructure size increases, the substrate attracts more argon atoms to form a non-evaporating liquid layer, improving heat transfer and promoting bubble nucleation.

A Natural Exposition and Concise Derivation of Geometric Optics Laws by Fermat Principle

With Fermat Principle, that is, the most fundamental principle in geometric optics, it is easy to obtain the three laws of geometric optics as well as the paraxial refraction and catoptric imaging laws. Making use of the principle and a simple mathematical relation, this paper expounds and derives the rectilinear propagation law of rays in the homogeneous medium, the optical reflection and refraction laws as well as the spherical reflection and refraction laws under the paraxial condition. Ulteriorly, with given character rules, the spherical dioptric and catoptric imaging formulas can be generalized into a unified formula separately. The elaboration and derivation in this paper reflect the simplicity of physics and reduce the error probability of complex trigonometric function operation, thus making it more easy for high school students or junior undergraduate and junior college students to accept and understand.

Three dimensional stabilization controller based on improved quaternion transformation for fixed-wing UAVs

To realize the three-dimensional stabilization control of fixed-wing unmanned aerial vehicles (UAVs), we analyze the nonholonomic characteristics, constraint non-integrability, and controllability of UAVs. To simplify the trigonometric function term in the dynamics of the UAV and avoid the singularity problem of the Euler angle in describing attitude, we use the quaternion theory to transform the dynamics of the UAV to avoid the complex trigonometric function derivation, which makes the dynamic matrix more concise. Based on this, a continuous periodic time-varying controller (CPTVC) is designed, and the effectiveness of the controller is proved using the homogeneous method. Finally, the results of the hardware in a loop simulation indicated that the exponential stability provided by the feedback controller can realize the three-dimensional stabilization of any initial position.

On some novel solution solutions to the generalized Schrödinger-Boussinesq equations for the interaction between complex short wave and real long wave envelope

This paper explores some novel solutions to the generalized Schrödinger-Boussinesq (gSBq) equations, which describe the interaction between complex short wave and real long wave envelope. In order to derive some novel complex hyperbolic and complex trigonometric function solutions, the sine-Gordon equation method (sGEM) is applied to the gSBq equations. Novel complex hyperbolic and trigonometric function solutions are expressed by the dark, bright, combo dark-bright, W-shaped, M-shaped, singular, combo singular, and periodic wave solutions. The accuracy of the explored solitons is examined under the back substitution to the corresponding equations via the symbolic computation software Maple. It is found from the back substitution outcomes that all soliton solutions satisfy the original equations. The proper significance of the explored outcomes is demonstrated by the three-dimensional (3D) and two-dimensional (2D) graphs, which are presented under the selection of particular values of the free parameters. All the combo-soliton (W-shaped, M-shaped, and periodic wave) solutions are found to be new for the interaction between complex short wave and real long wave envelope in laser physics that show the novelty of the study. Moreover, the applied method provides an efficient tool for exploring novel soliton solutions, and it overcomes the complexities of the solitary wave ansatz method.

Performance Analysis of Clock Gated Coordinate Rotation Digital Computer Based Least Mean Square Filter Design for Video Processing Applications

Design of a low power FIR filter has always been an area of intense research concern. Application of Co-ordinate Rotation digital computer (CORDIC) algorithm for designing a low power FIR filter is presented in this paper. CORDIC algorithm helps in the computation of complex trigonometric functions by using shift and adds mechanisms avoiding the need for a separate processor in hardware implementation. The challenge of noise interruption in real time video signal capturing is rejected greatly by using CORDIC LMS filter architecture. Also the area, power, and delay of the LMS filter are reduced to a good extent because of CORDIC multipliers. When compared to the previously discussed filter techniques in various works, the proposed technique shows 21.13% reduction in delay and 90.28% decrease in leakage power. Further improvement in dynamic power savings is obtained through clock gating technique applied to the sequential elements in the circuit, which reduces dynamic power dissipation by 70% from the previous works. The comparative study of benchmark filters with their specifications is presented with a detailed analysis of area, delay, leakage, and dynamic power. The video processing using CORDIC LMS filter is verified with MATLAB R 2021A software. The hardware implementation of the proposed architecture is done on a SPARTAN 3E FPGA kit with Xilinx ISE design suite interface. The design is also verified for delay, area, leakage, and dynamic power using CADENCE with GPDK 180 nm library.

Dynamic interaction of behaviors of time-fractional shallow water wave equation system

In this study, the traveling wave solutions for the time-fractional shallow water wave equation system, whose physical application is defined as the dynamics of water bodies in the ocean or seas, are investigated by [Formula: see text]-expansion method. The nonlinear fractional partial differential equation is transformed to the non-fractional ordinary differential equation with the use of a special wave transformation. In this special wave transformation, we consider the conformable fractional derivative operator to which the chain rule is applied. We obtain complex hyperbolic and complex trigonometric functions for the time-fractional shallow water wave equation system with the help of this technique. New traveling wave solutions are obtained for the special values given to the parameters in these complex hyperbolic and complex trigonometric functions, and the behavior of these solutions is examined with the help of 3D and 2D graphics.

Optical Solutions of the Date–Jimbo–Kashiwara–Miwa Equation via the Extended Direct Algebraic Method

In this study, the solutions of 2 + 1 -dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation are characterized, which can be used in mathematical physics to model water waves with low surface tension and long wavelengths. The integration scheme, namely, the extended direct algebraic method, is used to extract complex trigonometric, rational and hyperbolic functions. The complex-valued solutions represent traveling waves in different structures, such as bell-, V-, and W-shaped multiwaves. The results obtained in this article are novel and more general than those contained in the literature (Wang et al., 2014, Yuan et al., 2017, Pu and Hu 2019, Singh and Gupta 2018). Furthermore, the mechanical features and dynamical characteristics of the obtained solutions are demonstrated by three-dimensional graphics.

Exact solutions of the stochastic new coupled Konno-Oono equation

In this paper we consider the stochastic Konno-Oono equation, which is forced by multiplicative noise. In order to find exact solutions of stochastic nonlinear Konno-Oono equations, generalized G′G-expansion method are implemented. Different forms of traveling wave solutions such as complex hyperbolic function and complex trigonometric function have emerged with various parameters. The results obtained in this paper extend and improve some results that were previously obtained. Moreover, we plot 3D surfaces of analytical solutions obtained in this paper by using Matlab to illustrate the impact of multiplicative noise on the solutions of the Konno-Oono equation.

Real Terrain Simulation based on Point Cloud of Water Area and Land Area

The joint simulation of land and water integration plays an increasingly important role in surveying and mapping, water conservancy, and land resources supervision. Based on the theory of 3D coordinate transformation, this paper discusses the application of Rodrigues’s matrix and indirect adjustment model in the iterative calculation of point cloud registration parameters. In this method, anti-symmetric matrix elements are used to form a rotation matrix, which avoids the complex trigonometric function operation of traditional methods, and improves the efficiency of calculation while ensuring the accuracy, and can easily calculate the initial value of parameters. Experimental results demonstrate that the proposed method has the advantages of high accuracy, robustness, stability, and reliability.

A series of abundant new optical solitons to the conformable space-time fractional perturbed nonlinear Schrödinger equation

In this work, we are investigating a series of new optical soliton solutions to the perturbed nonlinear Schrödinger equation (PNLSE) having the form of kerr law nonlinearity with conformable space-time fractional. Thereby, two relevant integration tools known as new extended direct algebraic method and extended hyperbolic function method are applied to obtain varieties of optical soliton solutions. The series of soliton solutions with fractional derivative order obtained by these methods can be classified as complex trigonometric and hyperbolic functions as well as other elementary functions. Furthermore, conditions for validity of the obtained analytical solutions, graphical illustration (2-D, 3-D) point out the impact of the fractional-order used.

Analytical and numerical approaches to nerve impulse model of fractional‐order

AbstractWe consider a fractional‐order nerve impulse model which is known as FitzHugh–Nagumo (F–N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional‐order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave‐type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers.

Fixed Points, Symmetries, and Bounds for Basins of Attraction of Complex Trigonometric Functions

The dynamical systems of trigonometric functions are explored, with a focus on tz=tanz and the fractal image created by iterating the Newton map, Ftz, of tz. The basins of attraction created from iterating Ftz are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x- and y-axis symmetry of the Newton map and explore the nature of the fractal images.

Optical solitons with full nonlinearity for the conformable space-time fractional Fokas–Lenells equation

In this paper, the extended direct algebraic method is applied to investigate the complex trigonometric and hyperbolic function solutions, especially dark, singular, combined dark-bright, combined singular, combined dark-singular, combined bright-singular solitons and periodic-singular solutions are obtained of the conformable space-time fractional Fokas–Lenells equation. The existence criteria for these solutions are also presented. The obtained results demonstrate that proposed strategy is powerful, effective and straightforward to work out new solutions of various types of nonlinear fractional partial differential equations (FPDEs) in applied sciences. Furthermore, the graphical representation of the solutions is also presented.

Investigation of dynamics of nematicons in liquid crystals by extended sinh-Gordon equation expansion method

This paper secures new nematicons in liquid crystals from its governing equation by extended sinh-Gordon equation expansion method with the aid of symbolic computational package Maple. The two laws of nonlinearity, namely Kerr and parabolic laws are studied. As outcomes, some new explicit complex hyperbolic and complex trigonometric function solutions are derived regarding dark, bright, combined dark–bright, singular, combined singular optical, periodic wave, dipole soliton solutions and others. To show the real physical meaning of the derived results, three dimensional graphs with contours and two dimensional graphs are prepared under the choice of proper values of the arbitrary parameters. The derived solutions have been checked back into their corresponding equations with the symbolic computational package Maple. All the combined solutions are found to be new, which may play an important role in the context of liquid crystals. The applied method is found to provide a powerful tool for extracting exact solitary wave solutions and overcome the difficulties of the solitary wave ansatz method.

Can Trichromacy equal Tetrachromacy?

Although our distant ancestors once had the same visual sensor arrangement as extant birds and reptiles, colour vision in most mammals is very limited; essentially reduced to a single dimension of information produced by a pair of colour sensors, each with a fairly broad spectral response. Humans belong to a small subset of mammals that have re-developed some of the ability that was lost by their distant ancestors. They have done this by stretching the natural variation of the light absorbing pigments in one of the visual sensors, thereby adapting it toward a third colour sensor. It is generally assumed that due to the differences and limitations of this solution that colour perception in humans is fundamentally different to that of our distant relations; animals which have been able to continually extend and refine their colour vision over a period of more than two hundred million years. However, over that extended period of time the fundamentals of their colour vision have remained largely unchanged; a system of four visual sensors, each with a specific colour filter that systematically narrows the spectral response. In this paper we demonstrate that a three-sensor system with a broad spectral response (similar in spectral profile to that of the human colour sensors) can in certain conditions be equivalent to a four-sensor system (broadly similar to bird and reptile colour sensors). This suggests that human colour perception (sometimes referred to as a 'trichromatic' system) may be broadly equivalent to colour perception found in a variety of other animals such as birds or reptiles (whose colour vision is generally referred to as 'tetrachromatic'). The difference between a sensor system that uses four spectrally narrow colour sensors arranged into two opponent pairs and a three-sensor system is that the former relies on a simple difference measurement whereas the later requires the use of complex trigonometric functions - which are at least two orders of magnitude greater in computational complexity.

Modified extended exp-function method for a system of nonlinear partial differential equations defined by seismic sea waves

Nonlinear partial differential equations are the main area of focus for researchers and scientists doing research in applied mathematics. Finding solutions of these nonlinear partial differential equations had gained considerable importance over the last few decades. In this work, an analytical technique named extended exp-function method is introduced for finding archetype exact solutions of innovative nonlinear coupled Konno–Oono equation. Different types of travelling wave solutions, i.e. complex hyperbolic function and complex trigonometric function solutions, with numerous capricious parameters are revealed. Subsequently, by using Maple 16, we plot 2D and 3D surfaces of analytical solutions obtained in this article. The depiction of the technique is straight, useful and can be applied to other nonlinear systems of partial differential equations.

The new complex rational function prototype structures for the nonlinear Schrödinger‐inviscid burgers system

In this paper, a system representing the coupling between the nonlinear Schrödinger equation and the inviscid burgers equation in modeling the interactions between short and long waves in fluids has been investigated. The new algorithms, like improved tan(Φ(ξ)/2)‐expansion method and improved Bernoulli subequation function method, have been proposed. The proposed methods have been discussed comprehensively in this article. Using these methods, some new prototype for exact solutions such as complex exponential, complex hyperbolic, and complex trigonometric function solutions have been obtained for nonlinear Schrödinger‐inviscid burgers system. Through the present analysis, it has been established that, with the help of symbolic computation, these methods provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations.

New explicit soliton and other solutions for the conformable fractional Biswas–Milovic equation with Kerr and parabolic nonlinearity through an integration scheme

In this paper, we derived new explicit complex hyperbolic and complex trigonometric function solutions, especially dark, bright, combined dark–bright, singular, combined singular soliton and other soliton solutions from the conformable fractional Biswas–Milovic equation with Kerr law and Parabolic law nonlinearity that describes the long-distance optical communications and optical fibres for the solitons propagation. The reported results might play an important role in a nonlinear medium for explaining the physical meaning of the studied model. Therefore, it is also shown that the applied method provides a more powerful mathematical tool for investigating exact solitary wave solutions for many other nonlinear evolution equations in nonlinear medium.

New closed form soliton and other solutions of the Kundu–Eckhaus equation via the extended sinh-Gordon equation expansion method

In this paper, the extended sinh-Gordon equation expansion method (ShGEEM) is applied to look for new complex hyperbolic and complex trigonometric function solutions, especially dark, bright, combined dark–bright, singular, combined singular soliton and other soliton solutions are obtained of the Kundu–Eckhaus (KE) equation which demonstrates the physical explanation in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. By choosing randomly the special values of free parameters, the solitary wave solutions are shown different patterns. The reported results might play an important role in the quantum field theory and nonlinear optics for explaining the physical meaning of the studied model. The obtained results also demonstrate that the extended ShGEEM is more competent than the other applied method applied by authors in the literature. Therefore, it is shown that the applied method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations arises in mathematical physics and nonlinear optics.

Increase of resolution by use of microspheres related to complex Snell's law.

The increase of resolution by the use of microspheres is related to the use of evanescent waves satisfying complex Snell's law with complex trigonometric functions related to the incident and refracted angles, while the refractive indices are real. The evanescent waves are obtained in addition to initial propagating waves satisfying the ordinary Snell's law. The lateral spatial wave vectors of the evanescent waves, which include information on the object fine structures, are converted at the microsphere surface to smaller wave vectors. Due to the reduction in the magnitudes of the spatial wave vectors of the evanescent waves, they become propagating waves including the fine structures which are recovered in the image plane.